Regularity and Bernstein-type results for nonlocal minimal surfaces

We prove that, in every dimension, Lipschitz nonlocal minimal surfaces are smooth.
Also, we extend to the nonlocal setting a famous theorem of De Giorgi stating that
the validity of Bernstein's theorem in dimension $n+1$ is a consequence of the nonexistence of $n$-dimensional singular minimal cones in $\R^n$.